Andrew Mead. An Introduction to the Music of Milton
Babbitt. Princeton: Princeton University Press, 1994.
321pp.
Reviewed by Jason Eckardt
Milton Babbitt's compositions and writings stand as some of the most
dazzling and provocative contributions to post-World War II music and
scholarship. His music is among the most individual and vital of our time,
and his articles and teaching have played an important role in the establishment of composition and contemporary music theory as viable academic pursuits. Babbitt was also one of the pioneers OIf electronic music
composition, and several of his works in this medium are considered seminal. It therefore seems appropriate that his work be given the treatment it
deserves in an extended volume.
Much of the writing on Babbitt's music has concerned his compositional methods. An Introduction to the Music of Milton Babbitt continues this
tradition, and Andrew Mead writes on his subject with authority, lucidity,
and enthusiasm. The author's analysis and discussion of Babbitt's compositional practices are consistently decisive and articulate and will reward
careful study. The bibliography is extensive, and the discography and list
of compositions are thorough and helpful. Mead's discussions of the compositional factors and designs in Babbitt's music are excellent. His writing
is generally clearheaded and straightforward. Several works are considered, and many musical examples are provided. Mead has wisely chosen to
analyze, for the most part, works that have been recorded, so that the
reader may listen to works discussed in the text. For anyone interested in
the way Babbitt makes his music, this volume is a valuable resource.
My primary criticism of the book is that Mead's analyses, as fine as they
are in detailing Babbitt's compositiOinal techniques, often do not serve the
author's stated intention to "elucidate ways of hearing that will lead the listener to a greater appreciation of a rich and rewarding musical world"
(4). Mead proclaims that "the best introduction to Babbitt's music is to listen" (9), and he justifies his analyses, stating, "Our appreciation of ...
Babbitt's music ... depends on our ability to perceive the underlying
structure behind the local details, to follow events below the surface" (5).
Given that Mead's analyses illustrate the methods employed by Babbitt
in the composition of both the background and surface of his music, I
84
REVIEWS
85
assume that the "underlying structure" to which he refers in the previous
statement is Babbitt's compositional design for a given work.! Underlying
structure is probably inferred by assembling details of the musical surface
into meaningful patterns. But following those details may lead the listener
to infer different structures, ones that may have little or no relation to
Babbitt's compositional designs. Mead believes that "unless we attempt
to hear the ways [Babbitt's musical surfaces] reveal the underlying longrange motion ... that forms the lasting emotional drama of his compositions," the listener will only receive "incidental gratification" (4).
Although Mead and-I are in agreement that "[o]ne of the rewards of listening to and thinking about Milton Babbitt's music is that it constantly
offers up new interpretations" (202-03), the only "informed hearing[s]"
(3) that his analyses suggest are ones reliant on the perception of
Babbitt's compositional designs. I believe that an attentive listener can
have a rich and satisfYing hearing of Babbitt's music that yields both fascinating structures and strong emotional responses without any knowledge
of the background structures employed by the composer. My assertion assumes that the definition of an informed hearing of Babbitt's music is
open to debate; for Mead, it appears to be a closed issue.
Although he believes that a listener must engage in "a radical reinterpretation of the significance of virtually all our basic musical perceptions"
(11) to properly hear twelve-tone music, Mead spends frustratingly little
time discussing strategies for hearing structural aspects of this repertoire.
To comprehend twelve-tone music more clearly, Mead reasons, the listener must perceive pitch-class aggregates (the twelve pitch classes of the
chromatic scale) as they pass across the musical surface. He writes:
[G]iven a collection with a large number of different pitch classes,
each represented once, we can recognize ... whether or not any additional note represents a new pitch class. By interpreting the recurrence of a pitch class as a signal that we have crossed a boundary, we
can parse a highly chromatic undifferentiated musical surface into a
discrete series of large bundles of pitch classes that we might call perceptual aggregates. (12)
Key to Mead's position on hearing twelve-tone music is the perception of
repetition. By recognizing pitch-class repetition, the listener infers structural boundaries that segment the musical surface. Despite Babbitt's and
Schoenberg's predilection for pitch-class repetition within aggregates,
Mead maintains that this is a cornerstone of the perception of twelve-tone
music. 2
I question this assumption. As much recent writing on music cognition
suggests, pitch is only one parameter used by listeners to parse the surface
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CURRENT MUSICOLOGY
of an atonal work. 3 While pitch certainly plays a role in the listener's organizational strategies, factors such as timbre, dynamics, rhythm, register, articulation, temporal adjacency, and textural density are likely to be important, perhaps dominant, factors in perceptual segmentation. Since Babbitt
makes use of non-pitch parameters to articulate various pitch-class aggregate structures, it seems odd that the author does not acknowledge the
importance of these parameters. Furthermore, some pitches "present" on
the musical surface can sometimes elude a listener because of the acoustic
effect of "masking," whereby, due to the varying degrees of amplitude of
simultaneous sounds, lower-amplitude sounds become difficult or impossible to hear. 4 If certain pitches are unintelligible during audition, the listener's ability to hear aggregates, following Mead's strategy, would be
seriously impaired.
Any decent musical analysis will provide for the reader new ways of
thinking about the piece it analyzes and, therefore, influence the way in
which the reader hears that piece. Several of Mead's observations undoubtedly provide interesting things to listen for in Babbitt's compositions. Nevertheless, even the most careful and experienced listeners will
need to bend their ears to hear all the underlying structural detail described in Mead's extended analyses.
Chapter 1 offers a brief discussion ·of tonal and atonal music, followed
by a fifty-page primer on "Milton Babbitt's Compositional World." For the
reader with a moderate-to-thorough knowledge of these materials, this
chapter will serve as a reminder of the wide range of Babbitt's compositional technique and clarifY the terminology that appears in the following
chapters. However, for those unfamiliar with twelve-tone theory, this is
hardly an introduction. 5 Complex and abstract compositional procedures
are presented with few musical examples and are often explained hastily.
The second chapter, "Mapping Trichordal Pathways (1947-1960),"
looks at Babbitt's earliest mature works. Particular attention is paid to
Composition for Four Instruments (1948), Composition for Viola and Piano
(1950), the Woodwind Quartet (1953), and String Quartet No.2 (1954).
In this chapter, we are finally shown the depth of Babbitt's compositional
technique, as exemplified in his works. Certain invariants in Babbitt's
methods are discussed. Most important of these is the principle of maximal diversity (i.e., the systematic exhaustion of all possibilities in some
fixed domain).6 Serialism could be seen as possessing this property, since
the entire chromatic is cycled through before the repetition of a pitch
class. Babbitt extends this principle to domains other than pitch, such as
combinations of instruments, structures of the twelve-tone rows, and distributions of rows in individual compositions.
Mead's analysis of Composition for Four Instruments, for flute, clarinet, violin, and 'cello, provides a breathtaking example of Babbitt's compositional
REVIEWS
87
craft and invention. Although informed by the serial techniques of Schoenberg and Webern, Composition for Four Instruments exploits serial principles
in a highly idiosyncratic way.7 Perhaps most striking is Babbitt's construction and subsequent musical realization of a trichordal array.
As Mead shows in example 1, a trichordal array comprises four ordered
twelve-tone rows, each of which is partitioned into four groups of trichords; they are then superimposed on one another, so that each horizontal row of the array (or "lyne") and each vertical column forms an aggregate.
Each trichord of Composition for Four Instruments's array features an intervallic structure with a rising or falling interval of either three or four
semitones. Read horizontally, the top lyne of the array, shown in example
1, forms intervals within trichords of +3/-4, -4/+3, -3/+4, and +4/-3:
all the possible combinations of these interval pairs. Similarly, each columnar (vertical) aggregate is formed by four trichords, each of which represents one of the four intervallic ordering pairs. (The first column, from
top to bottom, reads +3/-4, -3/+4, -4/+3, and +4/-3.) The interval
pairs, because of these interval orderings, collectively present the four
"classical" transformations of the twelve-tone system: prime, inversion, retrograde, and retrograde inversion.
These trichords are registrally stratified in the opening clarinet solo of
Composition for Four Instruments (ex. 2), with the top two lynes of the array
in the high register and the bottom two lynes in the low register. Mead explains that Babbitt's array lyne-pairing choices are not arbitrary. Each
hexachord of a lyne may be combined with the hexachord of the other
lyne ofthe pair to create the aggregate. In this instance, if we combine the
first trichords of the lyne pairs, both result in the hexachord type [0 1 45
89].8 This is a harmonic contrast to the [0 1 2 345] hexachord type
found in the lyne aggregates (ex. 1). Most often, the trichords are not presented in their entirety but rather with some interpolation of pitch classes
from some other trichord from a different lyne. By the end of the first aggregate of Composition for Four Instruments (mm. 1-6, ending on F), we have
heard the completion of the first columnar aggregate, while each of the
lyne aggregates, remaining stratified in their own registers, is not completed until the fourth and final columnar aggregate is (mm. 13-16).
In Babbitt's compositions, abstract structures are realized on the musical surface and simultaneously embedded at deeper levels. As Mead observes, the conjoining of array lyne trichords, rather than their solo presentation, characterizes the opening of Composition for Four Instruments.
Example 3a illustrates the distinction between trichords that are intertwined and those presented alone, with regard to their register, temporal
placement, and their transformation type (which is derived from each
trichord's ordered interval pair, where the prime form is +4/ - 3). If we
88
CURRENT MUSICOLOGY
Example I (from Mead, p. 60): Trichordal array, CompositionJor Four Instruments (excerpt).
[0.1,2,3,4,5]
LYNE
PAIR
AGGREGATE
~
i
I
I
" I7P
~
[0,1 ,4,5,8,9]
"
"Q
oj
" I7Q
I
oj
P = prime, 17P = inverted prime at a transposition level of 7 semitones, Q = prime with trichord orderings reversed for each hexachord, 17Q = inverted prime at a transposition level
of 7 semi tones with trichord orderings reversed for each hexachord.
compare this chart to the distribution of instruments throughout the
eight sections of the piece (ex. 3b), it becomes clear that Babbitt is interpreting the same abstract pattern in two musical dimensions.
Mead discusses these points, and many others about Composition for Four
Instruments, with clarity. However, if his intention is to show us how "this
piece .. , illuminates a wide variety of Babbitt's compositional practices
and dramatically demonstrates their effect on our hearing of twelve-tone
music" (55-57), he neglects important issues. The above analysis is focused on the domain of pitch, exploring how pitch classes are stratified in
various registers and form trichordal interval patterns. Mead writes very
little about other parameters, many of which have a strong influence on
my hearing of Composition for Four Instruments. For example, Mead is
concerned with the perception of pitch-class repetition in the determination of "perceptual aggregates." It seems that the beginning of the third
aggregate of the clarinet solo should have particular salience for the listener. In m. 9, the clarinet completes its second aggregate on G-natural.
The clarinet's third aggregate begins with B3 on the last eighth note of
m. 9, with a dynamic marking of mezzo piano. As the third aggregate begins
with the same register, dynamic, and pitch class as the first sound in the
piece, I sugg'est that this is a clear indication of a structural boundary.
This boundary is powerfully reinforced by the two eighth-note rests that
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REVIEWS
Example 2: Opening measures, Composition for Four Instruments.
;PI
~~
mp
ff
mf
if~J ~i1 if
'------" ~ ff
f
r
ppp
--======
mf
# _______
Example 3a (from Mead, p. 61): Distribution of trichords, Composition for Four Instruments.
1+3
Partition:
High
Low
R
Rl
I
{
{ P
2+2
I
3+I
2+2
Rl
P
P
R
R
P
Rl
Rl
R
3+1
1+3
2+2
4+0
T6Rl T6P
T6R
T6R
T61
T6Rl
T6P
T6Rl T61
T6R
T61
T6P
T61
T6P
T6R
T6Rl
P = +4/-3
Example 3b (from Mead, p. 58): Distribution of instruments, Composition for Four Instruments.
Partition:
VIOLIN
CELLO
FLUTE
CLARINET
1+3
2+2
x
X
X
X
X
3+1
X
X
X
X
3+1
X
X
X
X
2+2
X
X
X
X
X
X
X
2+2
1+3
X
X
X
X
X
X
X
X
4+0
X
X
X
X
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CURRENT MUSICOLOGY
precede the B-natural, providing the longest silence heard thus far. Mead
does not acknowledge these factors.
The boundary between the first and second aggregates (mm. 6-7) is articulated in a different way. The first aggregate ends on F-natural (m. 6),
followed by a rest. On the next downbeat (m. 7), the C-sharp begins the
second aggregate. The dynamic of the previous three and a half measures
has been mezzo piano. With the beginning of the second aggregate, the dynamic environment changes. The C-sharp is sounded mezzo forte, and then
there is a sudden shift to piano on the B-natural, followed by a fortissimo
B-flat on the second half of the third beat (m. 7). This sudden flux of
dynamic levels offers a sharp contrast and seems to me an even stronger
indication of a boundary than the pitch-class repetition beginning in m. 7.
Furthermore, Babbitt writes a slur over the first six measures, and the articulation distinguishes the opening aggregate from those that follow.
Again, the boundary between aggregates is marked with a silence (an
eighth-note rest, m. 6). While an attentive listener might hear a boundary
between the D-natural and the F-natural in m. 6, since it represents the
longest rest heard thus far in the piece, the combination of a rest with the
following shift in dynamic and articulation beginning with C-sharp
strongly suggests a more significant boundary.
I have chosen to focus on only one of the works considered in chapter
2, which adequately demonstrates the rigor of Mead's analyses. Other
analyses in the chapter maintain a similar set of concerns, with particular
attention paid to the ways in which Babbitt creates new works through the
reinterpretation of the array structure used in Composition for Four
Instruments. These analyses are very detailed and the pace of presentation
is rapid; even a passing point will sometimes require close examination of
the examples and musical excerpts. One gets the impression that Mead
cannot write enough about Babbitt's music, and the author frequently
laments the lack of space available to expound on issues he raises.
The author's concern is more pronounced in chapter 3, "Expansion
and Consolidation (1961-1980)." Rather than attend to a few representative compositions, as he does in chapter 2, Mead discusses excerpts of several works to illustrate Babbitt's development of new compositional techniques. Because Babbitt's use of these new compositional tools becomes
even ·more particular to individual works, Mead does not want to shortchange the breadth of Babbitt's ideas in the interest of more extended
analyses of single pieces.
According to Mead, the "structural hallmark of Babbitt's compositional
practice in his second period is the all-partition array" (125). An allpartition array, like its trichordal counterpart, is a two-dimensional concatenation of lynes partitioned so that their individual segments, when
superimposed on other segments from different lynes of the array, create
REVIEWS
91
aggregates that are formed vertically, in columns. An array in which lynes
are partitioned in all possible ways, with respect to the number of pitch
classes in each partition, is an all-partition array (ex. 4). These lynes are
paired together in the array, and also become connected in particular
ways in Babbitt's music, as they were in Composition for Four Instruments.
Mead also discusses a major change in Babbitt's compositional technique: the development of the time-point system, a method of interpreting arrays in the rhythmic domain. While Babbitt had been using various
techniques to derive his rhythms from serial structures, the time-point system carried heavier implications for rhythmic formation in his work. 9 In
the time-point system, Babbitt interprets the integers of the array as attack
points in a finite temporal span, or modulus, which is divided into twelve
equal parts. For example, a simple modulus is a measure of 2, in which
each sixteenth note in the measure represents one of twelve time points.
If 0 is considered the first sixteenth note, then 5 falls on the sixth time
point. That time point in a modulus of 2 occurs on the sixth sixteenth
note of the measure, the second sixteenth note of the second beat. (The
time-point system only prescribes the placement of attacks, not duration.)
In Babbitt's works, these time points are coordinated in various ways with
pitch-class interpretations of the same array, or some transformation of it.
The time-point system provides Babbitt with a new and powerful way
of structuring surface rhythmic formation. However, Babbitt's treatment
of the dimensions of pitch and rhythm raises certain issues for Mead. Most
problematic is the reconciliation of the time-point modulus with octave
equivalence. By dividing the time-point modulus into twelve parts, with
each consecutive set of twelve time points beginning a new modulus,
Babbitt makes an analogy to pitch-class octave equivalence. Clearly, however, these are two domains so different that they hardly seem complementary. Babbitt himself is quick to point out that the number twelve has
absolutely no inherent significance in the temporal domain (Babbitt
1962:71-72).10 It is worth noting, as Mead does, that a pitch-class interpretation of the array in Babbitt's music is most often dependent on register
or timbre for the stratification oflyne pairs on the musical surface. Babbitt
turns to other parameters, usually dynamic, to connect time points in individual lynes and to distinguish lynes or lyne pairs from one another. It is
through these techniques of distribution of time points and pitch classes,
Mead argues, that Babbitt is able to realize rhythmic structures with the
same subtlety that he does his harmonic structures.
While critics have tried to reduce Babbitt's array composition to the
obligatory fulfillment of a series of a priori decisions, Mead's sections
"Composition of Details" and "A Brief Venture into Comparative Anatomy" convincingly present arguments to the contrary. Mead writes that
"arrays contain certain structural potentials that can be realized in a
92
CURRENT MUSICOLOGY
Example 4 (from Mead, p. 274): All-partition array,
T-P
P-C
Mod. Reg
5 2
67 79 4
8
high
0 8e
t 31
1 8
te
3
0 4976
3
6 9
954 2
7 e
2
8ge e6 6 0532
7 4
7
3 to 01
5
5
2
t
0 5
tl
32
4
4
76
4 9
8 e
1
e 6 89
It
6
3 3 t
0
2 7 54
4
5 81
732 0
5
low
8
9
6te 18
Post-Partitions (excerpt).
03
7245
6
2
e8
74
3t
3
59
e
5
01
1
2t
8e764
489
t8
6
81
3
t
9
e6
0152
304572
t19
986
Ie
50
e
t6
0237 74
9
T-P Mod. ~ Time-Point Modulus
p-c Reg. ~ Pitch-Class Register
number of different strategic solutions" (128) and illustrates this point by
showing us the ways in which Babbitt "composes out" his array structures.
Mead demonstrates how, by stratifying their materials in particular musical dimensions, Babbitt takes the same abstract array structures and uses
them to create remarkably different pieces.
Chapter 3 ends with two extended analyses, the first of which discusses
Post-Partitions (1966), one of Babbitt's most well-known compositions for
the piano. The opening of this work reveals much about Babbitt's middleperiod compositional practice. Post-Partitions begins with an outburst of
dyads, which exhaust the entire registral range and dynamic capacity of
the piano. The pitch and rhythmic material of the first measure is represented in the first column of Post-Partitions's array (ex. 4). In the pitch domain, Babbitt has stratified each lyne pair in register from high to low,
and the dyads that appear in the opening are composed of these array
lyne pitch classes. Each lyne in each pair is distinguished by articulation:
the top lyne of each pair (but not necessarily the highest in register) is
presented staccato, while the lower lyne is sustained. In the rhythmic domain, Babbitt casts each lyne pair in a different time-point modulus, from
thirty-second-note subdivisions to triplet eighth-note subdivisions. Each
time-point integer is distinguished by one of twelve dynamic levels from
JJJJJ to pppppll (ex. 5) .12, 13 Example 6 reproduces the first measure of PostPartitions, with time-point integers enclosed in circles. 14 Although this example is brief, it nevertheless brings to light the vigor of Babbitt's compositional techniques as described by Mead.
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REVIEWS
Example 5 (from Mead, p. 175): Time-point moduli, Post-Partitions.
Time points:
Dynamics:
f
2
0
1
0
1
0
1
2
3
4
5
6
7
8
9
fJfjJ
ppppp
pppp
ppp
pp
p
mp
mf
f
ff
3
4
5
6
7
8
6
7
9
t
I0
e
2
1
3
4
5
6
1
2
3
I
I
7
9
8
e
I0
1
8
9
t
t
2
e
fff ffff
3
4
5
6
7
8
2
3
4
5
8
9
t
e 10
4
5
7
6
elo
2
1
3 ..
7
I
0
I
7
2
1
4
3
I
7
6
5
7
8
9
eI 0
t
I
7
2
1
4
3
I
5
I
7
6
7
8
9
eI...
t
6
I
I
6
0
2
1
3
4
I
I
6
5
6
7
8
9
I
I
6
t
I
e
0
I
2
1
I
6
3
4
5
6
7 ...
5
~5~
0
2
1
3
~5~
4
5
6
7
~5~
8
9
e
t
~5~
I
0
2
1
3 .
4
0
1
2
3
4
5
6
7
8
9
t
0.1·
3
~3-
~3-
~3-
~3-
Again, Mead provides much valuable information too detailed to mention in this context. In his pursuit of deeply embedded structural connections, however, he sometimes loses sight of his intention to provide information that might contribute to a clearer hearing of Babbitt's music. For
example, his analysis of Post-Partitions ends with a discussion of the last
attack of the piece and the events that lead up to it. The final measure of
Post-Partitions (ex. 7) features a jJffJ attack on F-sharp, C-sharp, and F, over
an A-flat sustained from the previous measure. Mead connects these pitch
classes to the E found in the highest register of the penultimate measure
and to the E-flat that precedes it, also in this measure. Mead's goal is to extract the [0 2 3 4 5 7] hexachord type from the texture, since this hexachord underlies the formation of array lynes and surfaces throughout
the piece. Mead points out that all the pitches of this hexachord are
marked with loud dynamics, and the E-flat and E appear in the higher
registers of the penultimate measure. However, the E-flat and E appear simultaneously with their dyadic counterparts of G and B, respectively.
94
CURRENT MUSICOLOGY
Example 6: First aggregate, Post-Partitions, m. 1, with time-point annotations.
CD
pppp
CD
~5~
l-v-lt. ~ ~j d~J~.J
j
=90
se;:pre 8 higher
ff
-
eJ
CD
"
mf
CD
ppppp
r---3_
-.
®
~rlr
Jff
pp
pppp
CDppp
CD
•
p'
ffff
CD
p-
~~Jt
el
3
_u
_"lI_
r~
ffJff
(8)
J
I
mp
sempre 8 lower
®
~.
~.
i
i
f
ff.Iff
(8)
Example 7: Last two measures, Post-Partitions.
ppppp
r---
5,
~n
8;,-
p
~5----'
"
'~
ff
ffff
ffJff
~.
ppp
pp~
0;
'I
L--------7~
pppp
8--
If''··
p
REVIEWS
95
Furthermore, the B of the {E B} dyad is higher in register than the E-flat.
The dynamic level of the first {E B} dyad attack is ppppp, a distinctly lower
dynamic than the dynamics ranging from f to ffJJf associated with other
members of this hexachord type. Essentially, Mead expects the listener to
ignore the lower-register pitch classes of the {E~ G} and {E B} dyads, to disregard the ppppp dynamic level of the first attack of the {E B}dyad, and
then to connect the E-flat and E to the {F# C# A~ F} appearing in a different register in the last measure. Since the [0 2 3 45 7] hexachord type is
an important structural element in the piece, Mead may claim that the
hearing he proposes has been "prepared." Nonetheless, on a surface as
dense and restless as that of Post-Partitions, a hearing this specific to hexachordal associations may prove inaccessible for even the most careful
listener.
Over the past forty years, Babbitt's structural concerns have grown
more vast and elaborate, and his compositional techniques have developed accordingly. The fourth and final chapter, "The Grand Synthesis
(1981-) ," focuses on Babbitt's present compositional procedures, which
have produced perhaps the most complex and individual works of his career; they have also supported works of the greatest scope with regard to
their variety of surface figurations and details, underlying structural formations, duration, and instrumental forces.
Central to the composition of Babbitt's recent music is the superarray,
a structure created by layering arrays on top of one another. The superarray provides Babbitt with an even denser contrapuntal framework within
which to compositionally maneuver. Babbitt maintains careful control of
the superarray's presentation, using parametric definition (in the domains
of register, dynamic, timbre, and articulation) to distinguish the constituent arrays of the superarray, much in the way he distinguished lynes
and lyne pairs in his earlier compositions. Implicit in the superarray is the
duplication of pitch classes during the simultaneous presentation of multiple columnar aggregates. 15 Babbitt's composing-out of the superarray in
various works both highlights and obscures this property.
In example 8a Mead provides the superarray whose first composite column is realized in the opening of Mehr ''Du'' (1991), for soprano, viola,
and piano (ex. 8b). The two simultaneous aggregates of the first three
measures are assembled from lynes that are stratified in the lowest register
of the viola in the top array, and in the low voice, mid-high and mid-low
registers of the piano in the bottom array. Here, Babbitt avoids setting the
same pitch classes in rhythmic unison or in temporally adjacent positions.
The synthesis to which Mead refers in the title of his last chapter is the
fusion of two major concerns in Babbitt's previous periods: trichordal and
all-partition arrays. While some of his superarrays utilize a single array
96
CURRENT MUSICOLOGY
Example 8a (from Mead, p. 254): Superarray, Mehr "Du" (excerpt).
23elOt
suI pont.
pizzo
Top
Array high
I(e)
t9
754623
Va.
9
3
980tel
8
5
med
8
7e
562
e9tO
2431
e98t67
023154
low
4
Voice {
(high)
Voice {
(low)
Bottom
Array
6845
976
34021
5786t
ge
t
5
Pno.
~
low
7568
Ot21
7
9t
20el
et2
467
high
437
0
786
9
79t8
643
201
598
34
6t
3
402
8
Ie
t9
7
tel
980
54(6)23
0
351
te
104(2)35
e7
784653
986
5
I
Oe
2
REVIEWS
97
transform,ed through twelve-tone operations to produce derived arrays
that are then superimposed, others utilize non-derived constituent arrays,
often combining trichordal arrays common to the first period with allpartition arrays of the second. Trichordal formation, crucial to earlier
pieces, continues to occupy Babbitt in the form of the all-trichord row, a
twelve-tone row whose ordering contains overlapping partitions of nearly
all of the trichord types.l 6 Babbitt further integrates his former practices
by sometimes using these all-trichord rows as the lynes in all-partition arrays. These arrays are then themselves combined with other arrays to form
superarrays.
Perhaps because there is no "typical" Babbitt superarray composition,
Mead shies away from extended analyses, opting mostly for ones that last
no more than a few pages. Presumably, Mead is attempting to illustrate
the various ways in which Babbitt has reinterpreted a central structural
idea in his music. Yet, with ideas as complex as those presented here, it
would have been more rewarding to be guided through a significant portion of a superarray composition. Noticeably absent is any extensive discussion of Babbitt's recent time-point techniques, leaving the reader to
wonder how Babbitt has integrated new developments such as the superarray into the rhythmic domain.
In the brief analyses Mead does offer us, he details the structural richness and surface variety in Babbitt's recent music. Again, I am disappointed that Mead is so determined to offer us analyses outlining Babbitt's
compositional designs that many striking features of the music are
slighted or ignored. For example, in his discussion of Whirled Series (1987),
for saxophone and piano, Mead describes the end of the piece as a "ferocious conclusion [where] the final composite aggregate [is] composed out
over an extraordinary twenty-four bars" (226). The ending to this piece is
so unique that Mead obviously felt it was worth mentioning. But in light
of Mead's attempt to offer information that will aid in our hearing of
Babbitt's music, I am surprised that he does not discuss in greater depth
the sense of finality that is achieved by the suspension of the last composite aggregate. A cadence is generally thought of as a harmonic resolution
to a stable point of rest. While all of the music that precedes the conclusion of Whirled Series is characterized by a much faster aggregate turnover
rate, the ending obsessively reiterates the final composite pitch-class aggregates of the superarray over a considerable period. It thus sounds comparatively stable, setting the piece's harmonic motion at rest by settling on a
"cadential" aggregate group (ex. 9).
An Introduction to the Music of Milton Babbitt provides a meticulous and
comprehensive study of Babbitt's compositional techniques. As an introduction to Babbitt's music, this book will be a challenge for anyone not
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CURRENT MUSICOLOGY
Example 9: Final measures, Whirled Series.
~1'~
Alto Sax
(in E)
~
#~.
~~. ~~
~
~/l~
pp~
-==== f -=== f.f
,,~
) .""
f.f
ppf
~
r------
" - - 3----'
r?
Jj
oJ
,,,'
~~~~
(I;~ ~~~~
=-
'------- 6-------"
p
"-~
~7J:
possessing a strong background in twelve-tone theory and not somewhat
familiar with Babbitt's compositional techniques. As a guide to hearing
Babbitt's music, An Introduction frequently relies too heavily on the relationships and features found in the composer's structural designs. Mead
often disregards the more obvious surface characteristics that may lead a
listener to some of Babbitt's structural underpinnings, and he never examines ways structure might be aurally perceived outside of Babbitt's compositional designs.
To be fair, Mead remarks that his analyses only "scratch the surface"
(76, 202) or "pass over much of the richness of the composition's details"
(115). Because Mead believes that Babbitt's underlying compositional designs are "the source that animates the farthest reaches and ramifications
of the sounding surface" (5), perhaps he would argue that we are always
in some sense "hearing" Babbitt's designs since surface details are local
manifestations of deeper levels of large-scale structure. Nevertheless, the
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99
Example 9 (cont.)
.....,
~
7\
rJ~
~
.. ..
l J
17
::>-pp
~~ ~
~.
I
r
A~~~~fr
t1
1""1
.
p
-===17=
DIp
"
H*
~
)
I,....
In,".
'.~.
P\
pp
17
#~~~
p
17
:
""
"
~
~
":'
~
~J
,--- ------,
7:f________-.J~-.J
:::=-
17
~
"
~
P
_.J,-,",,'
",,~_.J_____________ ""~ ""
f
PP
~.~ .. ~
~-<
lI-J
17
:
P
f
PP
,-3----,
~~~ ~":' .~.~~J
r
~~
r.'\
#Hf3~'
~
)
8"'-----,
~
~
*
translation from background compositional structure to musical surface
realization may be too complex for the listener to explicitly infer an intricate compositional design such as an array in the degree of detail that
Mead describes in his analyses. What this book does show us is how Milton
Babbitt composes.
Notes
1. "Composition" here refers to the structural planning that produces a set of
constraints before the "composing" begins, as well as to the production of the musical score.
2. Mead acknowledges that "[i]n most twelve-tone music there are extensive
examples of pitch-class repetition within aggregates" (13), but he never makes
clear how the listener might distinguish internal pitch-class repetition within, as
opposed to between, aggregates.
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CURRENT MUSICOLOGY
3. Non-pitch parameters and their role in the perception of atonal music have
been explored by Deliege (1989), Krumhansl (1991), Clarke and Krumhansl
(1990), Tenney and Polansky (1980), and Nonken (1999), among others.
4. See Morris (1987:295); Bregman (1990:320-23,392).
5. Excellent resources for the terms discussed and notation used in Mead's
book may be found in Forte (1973), Rahn (1980), Morris (1987), and Straus
(1990).
6. For a detailed discussion of maximal diversity in selected compositions by
Babbitt, see Dubiel (1992).
7. Babbitt discusses the influence of Webern's Concerto for Nine Instruments on
the structure of the opening aggregate of Composition for Four Instruments in
Dembski and Straus (1987:24-28).
8. Pitch classes are notated in integers where C=O, C-sharp=l, D=2, etc. To
avoid confusion, the pitch-class integers 10 and 11 are notated as "t" and "e," respectively. Collection classes (i.e., pitch-class set types that are equivalent under
transposition and inversion) are notated in square brackets; curly braces denote
unordered collections that are not equivalent under transposition and inversion.
9. Babbitt's most notable earlier rhythmic practice was the use of duration
rows. Duration rows are built on orderings of durational values from 1 to 12 that
increase incrementally in size. For example, if the durational value of 1 is a sixteenth note, 2 would equal an eighth note, 3 would equal a dotted eighth note,
and so forth. Typically, duration rows are numerical translations of ordered pitchclass integers found in pitch-class rows.
10. It should further be stated that the number twelve has no inherent significance in any musical domain. While the octave in Western art music is divided into
twelve equidistant parts, this is still not an inherent quality of pitch.
11. Mead offers insightful comments with regard to Babbitt's dynamic markings, noting that the extremes of dynamic do not represent absolute levels of amplitude but rather "indicate inflections within a normal range of dynamics [so
that] we can hear in his music a series of contours of intensity that maintain their
identity under various sorts of transformations .... These notated contours reflect
the flexibility available with the dynamic range, a quality found in performances
of, if not precisely notated in, the more familiar repertoire" (176-77). However,
for a listener to be able to aurally distinguish, for example, time-point lynes by
hearing the incremental differences in the dynamics that stratifY these lynes, the
listener must make distinctions between absolute dynamic levels. Mead fails to address this contradiction.
12. Babbitt sometimes repeats a time point within a measure if the modulus is
shorter in duration than the measure, and the position of the time point within
the repeated modulus in the measure allows space for reiteration of the time
point. In m. 1 of Post-Partitions, time point 0 is repeated (as the fJJJf{A E} dyad attack
on the downbeat of beat four) as well as time point 2 (the pppp {B~ F} dyad attack
on the second triplet eighth note of beat three).
13. The dynamic stratification of individual time points coupled with the use of
different, simultaneous moduli in Post-Partitions is somewhat unusual in Babbitt's
REVIEWS
101
oeuvre. More often, time-point array lynes are cast in the same modulus, with each
lyne or lyne pair stratified by different dynamic levels.
14. In some cases, such as with time points 1 and 2, and 6 and 8, in m. 1 of PostPartitions, certain attacks, although from different moduli, occur simultaneously.
Babbitt specifies different dynamic levels for the upper and lower pitch classes of
the dyads that fall on simultaneous time points to distinguish the time-point lynes
from one another.
15. Pitch-class duplication within array columns also occurs in earlier works, in
which Babbitt has transformed the array in such a way that the lyne aggregates are
preserved but the array columns contain pitch-class duplications. These columnar
collections that contain pitch-class duplications, and are therefore less than twelve
pitch-class types, are referred to as weighted aggregates.
16. Mead suggests that Babbitt has omitted the [0,4,8] and [0,3,6] trichord
types because" [t] he former cannot be ordered to represent the four classical
transformations [prime, inversion, retrograde, and retrograde inversion] unambiguously, and the latter is the single trichord type that cannot generate an aggregate" (156).
References
Babbitt, Milton. 1962. Twelve-Tone Rhythmic Structure and the Electronic Medium.
Perspectives of New Music 1 (1): 49-79.
Bregman, Albert. 1990. Auditory Scene Analysis. Cambridge: MIT Press.
Clarke, Eric, and Carol Krumhansl. 1990. Perceiving Musical Time. Music Perception 7: 213-52.
Deliege, Irene. 1989. A Perceptual Approach to Contemporary Musical Forms.
Contemporary Music Review 4: 213-39.
Dembski, Stephen, and Joseph Straus, eds. 1987. Milton Babbitt: Words About Music.
Madison: University of Wisconsin Press.
Dubiel, Joseph. 1992. Three Essays on Milton Babbitt, Part III: The Animation of
Lists. Perspectives of New Music 30 (1): 82-123.
Forte, Allen. 1973. The Structure of Atonal Music. New Haven: Yale University Press.
Krumhansl, Carol. 1991. Memory for Musical Surface. Memory and Cognition 19:
401-11.
Morris, Robert. 1987. Composition with Pitch-Classes. New Haven: Yale University
Press.
Nonken, Marilyn. 1999. An Ecological Approach to Music Perception: Stimulus-Driven
Listening and the Complexity Repertoire. Ph.D. dissertation, Columbia University.
Rahn,John. 1980. Basic Atonal Theory. New York: Schirmer.
Straus, Joseph. 1990. Introduction to Post-Tonal Theory. Englewood Cliffs: PrenticeHall.
Tenney, James, and Larry Polansky. 1980. Temporal Gestalt Perception in Music.
Journal of Music Theory 24(2): 217-39.